I attack my opponent's Burned and Asleep Pokemon with Drowzee's "Sinister Suggestion". When my opponent flips for Burned and Asleep, are those flips automatically tails?
No. Those flips are between turns, not during your opponent's turn as required by the attack.
Source:
Furious Fists FAQ (2014-08-28), TPCi Rules Team (2014-08-28)